SET: Discover Math Behind a Simple Game

Cover design by: Brilliant Math / Article written by Katherine Li

SET is a game in which there is a deck of 81 cards (play at setwithfriends.com!). Each card has four properties: shape, number, color, and shading; and there are three possible choices for each property.

Property	Choice, that we label 0	Choice, that we label 1	Choice, that we label 2
Shape	Oval	Squiggle	Diamond
Number	Three	Two	One
Color	Red	Green	Purple
Shading	Solid	Blank	Lined

No two distinct cards have all the same choices for each property. The goal of the game is to find three cards such that for each property, the choices on each of the three cards are either all the same or all different. In the game, we call these three cards a SET.

Let’s play around with this mathematically: note that when we label our choice for each property as a 0, 1, or 2, we can represent each card as a 4-dimensional vector with all its components being a 0, 1, or 2. For example, in the above picture, the leftmost card could be represented by the vector

(1 0 0 0). Now that we’ve interpreted our cards mathematically, how can we interpret a SET?

Click here for the mathematical interpretation...

It turns out that this is extremely nice. Suppose vector A, vector B, and vector C are vectors corresponding to cards that form a SET. Let’s consider one property at a time: if the choices are all the same for that property, the corresponding components of A, B, and C for that property could be all 0, all 1, or all 2. If the choices are all the different, the corresponding components would be some permutation 0, 1, and 2. This is particularly nice because we notice that 0+0+0, 1+1+1, 2+2+2, and 0+1+2 are all divisible by 3. Since a SET requires the choices to be all the same or all different for each property, we have that the components of A+B+C are all divisible by 3. In other words, A+B+C ≡ 0 mod 3.